S Pugh is correct that the puzzle isn't impossible, but I share Lumen's concern about normal wordplay rather than wordplay indicating the grid entries. The grid gives minimal feedback and with clues as obstinate as this, I found the solving process like peeling off a stubborn sticky label. I'm all for having some hard puzzles - that's what the Listener is for - but I like them to be enjoyable too. The endgame was clever and reasonable straightforward, but I'm not sure it was worth the joyless slog it took to get there.
I hope this appeals to others more than it did to me.